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So far, we have encountered integer-type exponents. In this chapter, we will explore fractional exponents and their representation as radicals. A fractional exponent is an exponent on a variable or number that is not a whole number, like $x_{21}$.

A fractional exponent can be seen as the opposite of an integer exponent. For instance:

$4_{2}=16$

And:

$4_{21}=2$

You can see here that when the $2$ is in the numerator of the exponent, $12 $, the number is squared, making $4∗4=16$. Conversely, when the $2$ is in the denominator of the exponent, $21 $, the opposite happens where we end up with **the number that would make $4$ when squared.** That number is $2$ in this case: $2∗2=4$.

This is what we refer to as the **square root**. Maybe you already knew that the square root is the opposite of a squared number, but what we will discover is that a root, also known as a **radical**, is the opposite of any exponent.

We know that a **cubed** number is a number to the power of $3$. So, the opposite of this exponent would be a number to the power of $31 $. Much like the square root, the fractional exponent $31 $ can be represented by a radical and called the cube root. But, the cube root has a $3$ in front of the radical to specify the denominator of the fractional exponent. Take a look at the difference between the square root and cube root.

$x_{21}=x $

$x_{31}=3x $

You see that there is a $3$ in front of the radical for the cube root, though there is not a $2$ in front of the radical for the square root. It is a little inconsistent, but this is the convention for radicals. If there is no number in front of the radical, it is a square root. If there is a number in front of the radical, then that number is the denominator of the fractional exponent. Likewise, if there is a number in the denominator of a fractional exponent other than $1$ or $2$, that number goes on the **outside** of the radical, attached to the radical itself. This number in front of the radical is called the index.

Fractional exponents will not always have a $1$ in the numerator. When there is an integer in the numerator of a fractional exponent, that number goes **inside** the radical, attached to whatever is inside. Let’s see the example $x_{32}$:

$x_{32}=3x_{2} $

Here, the $2$ goes inside the radical attached to the $x$.

Now you have everything you need to convert a fractional exponent into a radical. You will occasionally be asked to do this because fractional exponents are confusing and undesirable in a math expression. You can consider radicals as the **simplified form** of fractional exponents.

You need to be comfortable converting from a fractional exponent to a radical and from a radical to a fractional exponent. These are core concepts frequently tested. So, we will go through several examples below.

What is $4x_{2} $ as an exponent?

The numerator of the fractional exponent is the number **inside** the radical, $2$. The denominator of the fractional exponent is the number attached to the outside of the radical, $4$.

So, the exponent form of this expression is $x_{42}$ which can be further simplified to $x_{21}$

What is the exponent form of $3x_{3} $?

(spoiler)

Numerator/inside: $3$

Denominator/outside: $3$

Fractional exponent form: $x_{33}=x_{1}=x$

What is $x_{25}$ expressed as a radical?

(spoiler)

Numerator/inside: $5$

Denominator/outside: $2$

Radical form: $x_{5} $

What is $x_{−32}$ expressed as a radical?

(spoiler)

The exponent is negative, so we need to put the whole radical under a fraction of $1$.

Numerator/inside: $2$

Denominator/outside: $3$

Radical form: $3x_{2} 1 $

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